A theoretical model of a smoothly bent, graphene-like sheet retains a sharp signature of a topological transition in its transverse energy response, even though the size of that response depends on microscopic details of the lattice. The calculation also finds that the leading transverse charge response is tied to the occupied band’s Chern number, which changes across the model’s two phases.
The work is an analytical and numerical study of a microscopic Haldane lattice model, not a measurement. The authors examined how a curvature-induced deformation could produce currents flowing sideways relative to the direction in which the sheet bends, focusing on both charge and energy currents.
Charge tracks the model’s topological phase
For the charge response, the leading gradient term contained only the contribution associated with the curvature-induced scalar potential. Its value was linked to the Chern number of the occupied band rather than to the orientation of individual nearest-neighbor bonds.
The model has a trivial regime with Chern number zero when the magnitude of the next-nearest-neighbor hopping is below its critical value, and a topological regime with Chern number plus or minus one above it. The transition occurs at two opposite critical hopping values set by the model’s staggered sublattice potential. In practical terms, the calculation predicts a change in the leading charge response when the model crosses that boundary.
Energy current exposes the lattice’s fine print
The energy response behaved differently. At first spatial-gradient order, it depended on how the deformation changed each of the three nearest-neighbor bonds. When all three bonds were given the same modulation, their contributions canceled and the calculated response vanished. The unequal, bond-resolved modulation generated by the curved profile instead left a finite response.
That distinction matters because it shows why the absolute energy response cannot be treated as a purely low-energy, universal quantity in the full lattice calculation. The result depends on the microscopic pattern of hopping changes and on the crystal orientation, even though the relevant three bond coefficients cancel when added without those bond-specific weights.
The authors constructed the energy-current operator from a local-energy continuity equation with an auxiliary gravitational field, then expanded the flat-lattice current vertex through third order in the momentum transferred along the bending direction. This let them compare the ordinary lattice response with the response of a linearized single-Dirac-cone model.
A jump survives at the transition
At third order, lattice corrections changed the absolute response away from the transition, but a clear discontinuity remained when the topological phase changed. Independent linear extrapolations of the two response branches to the positive transition point gave a jump of about 0.00672 in the stated units, close to the relativistic benchmark of about 0.00663.
The comparison with the single-Dirac-cone limit points in the same direction: as the mass parameter approached zero from either side, the normalized response approached plus or minus one-half, producing a unit jump at the transition. The lattice calculation therefore preserves the transition’s discontinuous fingerprint while allowing its absolute values away from the boundary to vary.
The quoted lattice jump is an extrapolated estimate rather than a directly evaluated value at the transition. That distinction is important when comparing the numerical result with the relativistic benchmark.
What the model included
The modeled sheet had a smooth pure-bending profile that varied along x while preserving translation along y. The calculation neglected changes in the lengths of the nearest-neighbor carbon-carbon bonds, so it focused on the prescribed hopping modulation and the associated scalar potential within that approximation.
For one numerical bending example, the profile used an amplitude of 3.7 nanometers and a width of 2.5 nanometers. The nearest-neighbor hopping was set to 2.66 electronvolts, and the sigma-bond hopping parameter to 6.38 electronvolts.
The study’s decomposition found no transverse energy-current contribution from the curvature-induced scalar potential at any gradient order examined. In the authors’ interpretation, the calculated transverse energy current therefore came entirely from the nearest-neighbor hopping modulation.
The document is an arXiv preprint, version one, dated 28 August 2026. The authors acknowledged support from Korea NRF under grant RS-2026-25492880.
Paper data and sources
Original title: Topological signatures in the curvature-induced energy response
Authors: Jaehyeok Lee, Iuegyun Hong, Jinhong Park
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-28
DOI: Not available
Original paper · Full text